Answer:
d ⇒ di^(21-22) = -di, facing right from where it was; order doesn't matter
Step-by-step explanation:
For direction defined as 1, i, -1, -i, a left turn is accomplished by multiplying by i. Then "L" left turns will be multiplication by i^L.
A right turn is equivalent to multiplication by -i, so R right turns will be multiplication by (-i)^R.
We know that turning left and turning right are inverse operations of each other, so we can also write a right turn as multiplication by i^(-1). Then R right turns will be equivalent to multiplication by i^(-R).
We know from the commutative and associative laws of multiplication that the order does not matter. So, L left turns and R right turns will be equivalent to multiplication by ...
(i^L)(i^-R) = i^(L -R)
Hence 21 left turns and 22 right turns is equivalent to a turn of ...
i^(21-22) = i^-1 . . . . one right turn
From an initial facing direction of 'd', the final facing direction after these 43 turns is -di.
